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302        XIV    INTEGRATION IN  LOCALLY COMPACT GROUPS

sequence (mj(Y(aj)fj  //)). (It is enough to prove that if the sequence of functions
(%(Y(^)/} ~~ /})) converges to a function F, then D*/ exists and is equal to F. Fix
a neighborhood W of e such that ||y(z)F  F|| < Jfi for all z e W, then fix a real
number 8 > 0 such that a$ e W for 0 ^ k < 8 w^ . Now show that for large y, and
k<Smj, we have

and deduce that ||r-1(YWO)/'-/) - F|| ^ e for 0 g r < S.)

In particular, if the sequence (<2jrmj]) tends to e for all r e R, then the sequence
OW/(Y(J)./}  /})) tends uniformly to 0.

14. (a) Consider the sequence (gt) constructed in Problem 1 l(c). Show that the sequence
of numbers ///*/ is bounded above. (Argue by contradiction, by supposing that there
exists a subsequence (/(#))* N sucn tnat ^m ^w/'W = 0. Reduce to the case where

ft-* 00

the sequence (g^)) converges uniformly to a limit g in G, and there exists a sequence
of points cf(fc) e U"(^) converging to an element c ^ e. Then there exists an element
flic*) e Uf<*) such that

Now use the hypothesis lim ntwlf(k) = 0, Problem 7 of Section 12.9, and the definition

*-KX>

of the Uj to deduce that for each r e R the sequence (^JJ<fc)3) tends to e. Using Problem
13(b), show that this implies that \\v(c)gg\\ = 0, contrary to the fact that g is
adequate.)

(b) Show that if g is the uniform limit of a subsequence of the sequence (g{\ then
g is X-differentiable for every one-parameter subgroup X. (Using (a) and Problem
ll(c), show that the functions i(t(X(\li))gi  gi) form a uniformly bounded and
uniformly equicontinuous set, and apply Problem 13(b).)

15.    Let G be a unimodular locally compact group and ft a Haar measure on G. Show that
a function /e -^cK^ j^) ^s equal almost everywhere to a function which is uniformly
continuous with respect to a right-invariant distance on G if and only if the function
s i * y(s)f, with values in ^c(^> /?) is continuous at the point e. (To show that the
condition is sufficient, consider a sequence (un) of functions belonging to #" +(G)

such that Supp(wn)c vn (notation  of (14.11.2)) and   | un(x) df$(x) = 1. Prove that

Noo(wn *//) tends to 0 with I///, by using (13.17.1) and the Lebesgue-Fubini
theorem.)

16.    If p. is any bounded measure on R, the integral

is defined for every z e C such that J>'z ^ 0, and is an analytic function of z in each of
the half-planes ,/z > 0, Jz < 0. It is analytic also at all x 6 R which do not lie in the
support of fji. The function F^ is called the Stieltjes transform of ju.
(a) Let x0 e R. Suppose that the restriction of /x to an open neighborhood V of 0
has a continuous density g with respect to Lebesgue measure on V. Show that as y
tends to 0 through positive values, >(FM(x0 4- /) tends to a limit equal to 7rg(x0).d. Then, for each neighborhood U of e, there exists a number e > 0 such that